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Degenerate critical points, homotopy indices and Morse inequalities. II. (English) Zbl 0631.58001

The author studies the critical points of functions on \(R^ n\). In particular, he answers a number of questions from part I [ibid. 350, 1-22 (1984; Zbl 0525.58012)]. Firstly it is shown that if an isolated critical point has trivial cohomology for coefficients in Z, then it has trivial homotopy index. Secondly, we study how the coefficient ring affects the cohomology of an isolated critical point. Thirdly, we show how the special neighbourhood constructed in our earlier paper can be used to give a new proof of the known lower estimates involving cuplength for the number of critical points of a function on a compact manifold. At the same time, we obtain information on the cohomology of the critical points we find.
Fourthly, we obtain information on the cohomology of the critical point obtained in Rabinowitz’s saddle point theorem and much more precise information in the case of a mountain pass point. Finally, we discuss removability of critical points.
More recently, we have considerably improved the results in the previous paragraph [see Degenerate critical points, homotopy indices and Morse inequalities, III, Centre Math. Anal., Rep. Aust. Nat. Univ. (1987)]. We have obtained a necessary and sufficient condition for a critical point on \(R^ n\) (where \(6\leq n)\) to be removable and have obtained information on the cohomology of the critical point in some “dual” variational principles and also in infinite-dimensional cases.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Citations:

Zbl 0525.58012
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